A two-dimensional polynomial mapping with a wandering Fatou component

Abstract

We show that there exist polynomial endomorphisms of C 2 , possessing a wandering Fatou component. These mappings are polynomial skewproducts, and can be chosen to extend holomorphically of P 2 (C). We also find real examples with wandering domains in R 2 . The proof relies on parabolic implosion techniques, and is based on an original idea of M. Lyubich.

Dynamics of transcendental Hénon maps

“…If z ∈ R 2n , then we have that Re n−1 i=0 f i (z) ≥ n−1 i=0 b i and Re f n (z) ≥ b n , and thus by (1) both H n (z) and H n−1 (z) belong to the ball of radius 1 2 n+1 centered at ∞.…”

“…, and thus by (1) both H n (z) and H n−1 (z) belong to the ball of radius 1 2 n+1 centered at ∞. If z ∈ R 2n , then we have that Re n−1 i=0 f i (z) ≥ n−1 i=0 b i and Re f n (z) ≥ b n , and thus by (1) both H n (z) and H n−1 (z) belong to the ball of radius 1 2 n+1 centered at ∞.…”

“…Let s n ≥ 0 be an increasing sequence of real numbers converging to 1 2 . Let α n ≤ π 2 be a decreasing sequence converging to 0.…”

“…Recall that wandering domains are known not to exist for one-dimensional polynomials and rational maps [30], but they do arise for transcendental maps (see for example [7]). In higher dimensions it is known that wandering domains can occur for holomorphic automorphisms of C 2 [20] and for polynomial maps [1], but whether polynomial Hénon maps can have wandering domains remains an open question. We will consider two types of wandering Fatou components, each with known analogues in the one-dimensional setting.…”

“…The dynamical behavior can be restricted even further by considering transcendental Hénon maps whose map f has a given order of growth. For example, if the order of growth is smaller than 1 2 , then Fix(F k ) is discrete for all k ≥ 1.…”